(x+1)(x-2)(x-4) 0

Jasonbradley

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Jun 16, 2024

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Solving the Inequality: (x+1)(x-2)(x-4) > 0

This article will guide you through solving the inequality (x+1)(x-2)(x-4) > 0.

Understanding the Problem

The inequality represents a polynomial function that is greater than zero. We need to find the values of x for which this function is positive.

Steps to Solve

1. Find the roots: The roots of the polynomial are the values of x where the function equals zero. In this case, the roots are:

   • x = -1
   • x = 2
   • x = 4

2. Create a sign table: We can use a sign table to track the sign of the polynomial in different intervals. The table will have the roots of the polynomial as dividing points and the intervals between them.

   Interval	x + 1	x - 2	x - 4	(x+1)(x-2)(x-4)
   x < -1	-	-	-	-
   -1 < x < 2	+	-	-	+
   2 < x < 4	+	+	-	-
   x > 4	+	+	+	+

3. Determine the solution: We want the intervals where the polynomial is greater than zero (positive). From the sign table, we see that the polynomial is positive when:

   • -1 < x < 2
   • x > 4

Solution

Therefore, the solution to the inequality (x+1)(x-2)(x-4) > 0 is:

x ∈ (-1, 2) ∪ (4, ∞)

This can be interpreted as "x is greater than -1 but less than 2, or x is greater than 4".

Visual Representation

You can also visualize the solution by plotting the graph of the polynomial function. The solution corresponds to the intervals where the graph lies above the x-axis.

This approach to solving inequalities involving polynomials is useful for understanding the behavior of the function and finding the range of values for which it satisfies the given condition.